Abstract

A landmark in the development of quantum electrodynamics was the discovery that emission–reabsorption of virtual photons modifies the value of energy levels in an atom from those computed by using Dirac’s equation (Lamb shift). An early result of statistical quantum electrodynamics was that the exchange of virtual photons in an ensemble of identical atoms leads as well to a change in the frequency of the radiation emitted from this system (cooperative Lamb shift). Dicke’s discovery that coherence effects lead to the shortening of the emission lifetime from a small sample by a factor equal to the number of atoms in the ensemble (superradiance or cooperative decay rate) was an early landmark in quantum optics. Both cooperative decay rate and cooperative Lamb shift were shown to have the same physical origin—the exchange of virtual photons, a process described by the Lienard–Wiechert dipole–dipole interaction. This effective potential is the kernel of the integral equation describing the dynamics of the system. This complex long-range kernel gives, for both cooperative quantities, strong dependence on the geometry of the atomic cloud. I summarize the known expressions for the initial cooperative decay rate and the cooperative Lamb shift in different geometries. The results for both the scalar photon and the vector photon (electrodynamics) theories for experimentally realizable systems of either uniform or phased polarization are given.

© 2012 OSA

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  1. E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
    [CrossRef]
  2. W. E. Lamb and R. C. Retherford, “Fine structure of the atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
    [CrossRef]
  3. H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
    [CrossRef]
  4. R. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [CrossRef]
  5. V. M. Fain, “On the theory of the coherent spontaneous emission,” Sov. Phys. JETP 9, 562–565 (1959).
  6. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
    [CrossRef]
  7. J. T. Manassah, “Statistical quantum electrodynamics of resonant atoms,” Phys. Rep. 101, 359–427(1983).
    [CrossRef]
  8. M. J. Stephen, “First order dispersion forces,” J. Chem. Phys. 40, 669–673 (1964).
    [CrossRef]
  9. R. H. Lehmberg, “Radiation from an N-atom system. 1. general formalism,” Phys. Rev. A 2, 883–888 (1970).
    [CrossRef]
  10. R. H. Lehmberg, “Radiation from an N-atom system. 2. emission from a pair of atoms,” Phys. Rev. A 2, 889–896 (1970).
    [CrossRef]
  11. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
    [CrossRef]
  12. R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
    [CrossRef] [PubMed]
  13. R. Roehlsberger, “The collective Lamb shift in nuclear γ-ray superradiance,” J. Mod. Opt. 57, 1979–1992 (2010).
    [CrossRef]
  14. W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
    [CrossRef] [PubMed]
  15. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
    [CrossRef] [PubMed]
  16. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
    [CrossRef]
  17. R. Friedberg and J. T. Manassah, “Eigenfunctions and eigenvalues in superradiance with x–y translational symmetry,” Phys. Lett. A 372, 2787–2801 (2008).
    [CrossRef]
  18. R. Friedberg and J. T. Manassah, “Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory,” Phys. Lett. A 372, 2514–2521 (2008).
    [CrossRef]
  19. A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
    [CrossRef]
  20. J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
    [CrossRef] [PubMed]
  21. S. Prasad and R. J. Glauber, “Coherent radiation by a spherical medium of resonant atoms,” Phys. Rev. A 82, 063805 (2010).
    [CrossRef]
  22. R. Friedberg and J. T. Manassah, “The decay dynamics of a slab of two-level atoms excited by an ultrashort resonant pulse,” Opt. Commun. 281, 3755–3761 (2008).
    [CrossRef]
  23. R. Friedberg and J. T. Manassah, “Time-dependent directionality of cooperative emission after short pulse excitation,” Opt. Commun. 281, 4391–4397 (2008).
    [CrossRef]
  24. R. Friedberg and J. T. Manassah, “Electromagnetic decay modes in a spherical sample of two-level atoms,” Phys. Lett. A 372, 6833–6842 (2008).
    [CrossRef]
  25. R. Friedberg and J. T. Manassah, “The dynamical CLS in a system of two-level atoms in a slab-geometry,” Phys. Lett. A 373, 3423–3429 (2009).
    [CrossRef]
  26. R. Friedberg and J. T. Manassah, “The metamorphosis in the emission angular profile from an inverted two-level atom system,” Opt. Commun. 282, 3089–3099 (2009).
    [CrossRef]
  27. J. T. Manassah, “The metastable states in the cooperative emission from a system of two-level atoms in a sphere,” Laser Phys. 20, 1397–1403 (2010).
    [CrossRef]
  28. R. Friedberg and J. T. Manassah, “Electromagnetic modes of an infinite cylindrical sample of two-level atoms,” J. Math. Phys. 52, 042107 (2011).
    [CrossRef]
  29. R. Friedberg and J. T. Manassah, “Non-Dicke decay in a small spherical sample with radially varying density,” Phys. Rev A 85, 013834 (2012).
    [CrossRef]
  30. A. Svidzinsky and M. O. Scully, “Evolution of collective N-atom states in single photon superradiance: effect of virtual Lamb shift processes,” Opt. Commun. 282, 2894–2897 (2009).
    [CrossRef]
  31. R. Friedberg, “Refinement of a formula for decay after weak coherent excitation of a sphere,” Annal. Phys. 325, 345–358 (2010).
    [CrossRef]
  32. R. Friedberg and J. T. Manassah, “The Eikonal-SVEAS analytic closed forms for single photon superradiance,” Laser Phys. 20, 250–258 (2010).
    [CrossRef]
  33. R. Friedberg, “Nonradiative transfer of excitation in coherent decay from a Gaussian atomic distribution,” J. Phys. B 44, 175505 (2011).
    [CrossRef]
  34. R. Friedberg and J. T. Manassah, “The CLS in an ellipsoid,” Phys. Rev. A 81, 063822 (2010).
    [CrossRef]
  35. R. Friedberg and J. T. Manassah, “Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state,” Phys. Rev. A 81, 043845 (2010).
    [CrossRef]
  36. J. T. Manassah, “Giant cooperative Lamb shift in a density-modulated slab of two-level atoms,” Phys. Lett. A 374, 1985–1988 (2010).
    [CrossRef]
  37. J. T. Manassah, “The dynamical cooperative Lamb shift in a system of two-level atoms in a sphere in the scalar photon theory,” Laser Phys. 20, 259–269 (2010).
    [CrossRef]
  38. A. Svidzinsky and M. O. Scully, “On the evolution of N-atom state prepared by absorption of a single photon,” Opt. Commun. 283, 753–757 (2010).
    [CrossRef]
  39. R. Friedberg and J. T. Manassah, “Analytic expressions for the initial cooperative decay rate and cooperative Lamb shift for a spherical sample of two-level atoms,” Phys. Lett. A 374, 1648–1659 (2010).
    [CrossRef]
  40. R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
    [CrossRef]
  41. D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
    [CrossRef]
  42. R. Friedberg and S. Hartmann, “Pulse induced radiation in the linear regime,” Opt. Commun. 2, 301–304 (1970).
    [CrossRef]
  43. J. T. Manassah, “Emission from a slab of resonant two-level atoms induced by a delta-pulse excitation,” Laser Phys. 22, 559 (2012).
    [CrossRef]
  44. J. T. Manassah, “Analytic approximate expression for the spectral distribution of the emission from a slab of resonant two-level atoms prepared by an ultrashort δ pulse,” Phys. Rev. A 85, 055801 (2012).
    [CrossRef]
  45. J. T. Manassah, “The Purcell–Dicke effect in the emission from a coated small sphere of resonant atoms placed inside a matrix cavity,” Laser Phys. 22, 738–744 (2012).
    [CrossRef]
  46. A. Svidzinsky, “Nonlocal effects in single photon superradiance,” Phys. Rev. A 85, 013821 (2012).
    [CrossRef]
  47. V. I. Yukalov and E. P. Yukalova, “Possibility of superradiance by magnetic nanoclusters,” Laser Phys. Lett. 8, 804–813 (2011); and references to earlier work cited therein.
    [CrossRef]

2012 (6)

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

R. Friedberg and J. T. Manassah, “Non-Dicke decay in a small spherical sample with radially varying density,” Phys. Rev A 85, 013834 (2012).
[CrossRef]

J. T. Manassah, “Emission from a slab of resonant two-level atoms induced by a delta-pulse excitation,” Laser Phys. 22, 559 (2012).
[CrossRef]

J. T. Manassah, “Analytic approximate expression for the spectral distribution of the emission from a slab of resonant two-level atoms prepared by an ultrashort δ pulse,” Phys. Rev. A 85, 055801 (2012).
[CrossRef]

J. T. Manassah, “The Purcell–Dicke effect in the emission from a coated small sphere of resonant atoms placed inside a matrix cavity,” Laser Phys. 22, 738–744 (2012).
[CrossRef]

A. Svidzinsky, “Nonlocal effects in single photon superradiance,” Phys. Rev. A 85, 013821 (2012).
[CrossRef]

2011 (4)

V. I. Yukalov and E. P. Yukalova, “Possibility of superradiance by magnetic nanoclusters,” Laser Phys. Lett. 8, 804–813 (2011); and references to earlier work cited therein.
[CrossRef]

R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic modes of an infinite cylindrical sample of two-level atoms,” J. Math. Phys. 52, 042107 (2011).
[CrossRef]

R. Friedberg, “Nonradiative transfer of excitation in coherent decay from a Gaussian atomic distribution,” J. Phys. B 44, 175505 (2011).
[CrossRef]

2010 (13)

R. Friedberg and J. T. Manassah, “The CLS in an ellipsoid,” Phys. Rev. A 81, 063822 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state,” Phys. Rev. A 81, 043845 (2010).
[CrossRef]

J. T. Manassah, “Giant cooperative Lamb shift in a density-modulated slab of two-level atoms,” Phys. Lett. A 374, 1985–1988 (2010).
[CrossRef]

J. T. Manassah, “The dynamical cooperative Lamb shift in a system of two-level atoms in a sphere in the scalar photon theory,” Laser Phys. 20, 259–269 (2010).
[CrossRef]

A. Svidzinsky and M. O. Scully, “On the evolution of N-atom state prepared by absorption of a single photon,” Opt. Commun. 283, 753–757 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Analytic expressions for the initial cooperative decay rate and cooperative Lamb shift for a spherical sample of two-level atoms,” Phys. Lett. A 374, 1648–1659 (2010).
[CrossRef]

J. T. Manassah, “The metastable states in the cooperative emission from a system of two-level atoms in a sphere,” Laser Phys. 20, 1397–1403 (2010).
[CrossRef]

S. Prasad and R. J. Glauber, “Coherent radiation by a spherical medium of resonant atoms,” Phys. Rev. A 82, 063805 (2010).
[CrossRef]

A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

R. Roehlsberger, “The collective Lamb shift in nuclear γ-ray superradiance,” J. Mod. Opt. 57, 1979–1992 (2010).
[CrossRef]

R. Friedberg, “Refinement of a formula for decay after weak coherent excitation of a sphere,” Annal. Phys. 325, 345–358 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The Eikonal-SVEAS analytic closed forms for single photon superradiance,” Laser Phys. 20, 250–258 (2010).
[CrossRef]

2009 (3)

R. Friedberg and J. T. Manassah, “The dynamical CLS in a system of two-level atoms in a slab-geometry,” Phys. Lett. A 373, 3423–3429 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “The metamorphosis in the emission angular profile from an inverted two-level atom system,” Opt. Commun. 282, 3089–3099 (2009).
[CrossRef]

A. Svidzinsky and M. O. Scully, “Evolution of collective N-atom states in single photon superradiance: effect of virtual Lamb shift processes,” Opt. Commun. 282, 2894–2897 (2009).
[CrossRef]

2008 (5)

R. Friedberg and J. T. Manassah, “The decay dynamics of a slab of two-level atoms excited by an ultrashort resonant pulse,” Opt. Commun. 281, 3755–3761 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Time-dependent directionality of cooperative emission after short pulse excitation,” Opt. Commun. 281, 4391–4397 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic decay modes in a spherical sample of two-level atoms,” Phys. Lett. A 372, 6833–6842 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Eigenfunctions and eigenvalues in superradiance with x–y translational symmetry,” Phys. Lett. A 372, 2787–2801 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory,” Phys. Lett. A 372, 2514–2521 (2008).
[CrossRef]

1990 (1)

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

1989 (2)

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
[CrossRef] [PubMed]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
[CrossRef]

1983 (1)

J. T. Manassah, “Statistical quantum electrodynamics of resonant atoms,” Phys. Rep. 101, 359–427(1983).
[CrossRef]

1973 (2)

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
[CrossRef]

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

1970 (3)

R. H. Lehmberg, “Radiation from an N-atom system. 1. general formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. 2. emission from a pair of atoms,” Phys. Rev. A 2, 889–896 (1970).
[CrossRef]

R. Friedberg and S. Hartmann, “Pulse induced radiation in the linear regime,” Opt. Commun. 2, 301–304 (1970).
[CrossRef]

1969 (1)

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[CrossRef]

1964 (1)

M. J. Stephen, “First order dispersion forces,” J. Chem. Phys. 40, 669–673 (1964).
[CrossRef]

1959 (1)

V. M. Fain, “On the theory of the coherent spontaneous emission,” Sov. Phys. JETP 9, 562–565 (1959).

1954 (1)

R. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

1947 (2)

W. E. Lamb and R. C. Retherford, “Fine structure of the atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[CrossRef]

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[CrossRef]

1932 (1)

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
[CrossRef]

Adams, C. S.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Bethe, H. A.

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[CrossRef]

Burnham, D. C.

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[CrossRef]

Chang, J. T.

A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

Chiao, R. Y.

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[CrossRef]

Couet, S.

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

Datskou, I.

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

Dicke, R.

R. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Fain, V. M.

V. M. Fain, “On the theory of the coherent spontaneous emission,” Sov. Phys. JETP 9, 562–565 (1959).

Feld, M. S.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Fermi, E.

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
[CrossRef]

Friedberg, R.

R. Friedberg and J. T. Manassah, “Non-Dicke decay in a small spherical sample with radially varying density,” Phys. Rev A 85, 013834 (2012).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic modes of an infinite cylindrical sample of two-level atoms,” J. Math. Phys. 52, 042107 (2011).
[CrossRef]

R. Friedberg, “Nonradiative transfer of excitation in coherent decay from a Gaussian atomic distribution,” J. Phys. B 44, 175505 (2011).
[CrossRef]

R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
[CrossRef]

R. Friedberg and J. T. Manassah, “Analytic expressions for the initial cooperative decay rate and cooperative Lamb shift for a spherical sample of two-level atoms,” Phys. Lett. A 374, 1648–1659 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The CLS in an ellipsoid,” Phys. Rev. A 81, 063822 (2010).
[CrossRef]

R. Friedberg, “Refinement of a formula for decay after weak coherent excitation of a sphere,” Annal. Phys. 325, 345–358 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The Eikonal-SVEAS analytic closed forms for single photon superradiance,” Laser Phys. 20, 250–258 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state,” Phys. Rev. A 81, 043845 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The metamorphosis in the emission angular profile from an inverted two-level atom system,” Opt. Commun. 282, 3089–3099 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “The dynamical CLS in a system of two-level atoms in a slab-geometry,” Phys. Lett. A 373, 3423–3429 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic decay modes in a spherical sample of two-level atoms,” Phys. Lett. A 372, 6833–6842 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “The decay dynamics of a slab of two-level atoms excited by an ultrashort resonant pulse,” Opt. Commun. 281, 3755–3761 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Time-dependent directionality of cooperative emission after short pulse excitation,” Opt. Commun. 281, 4391–4397 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory,” Phys. Lett. A 372, 2514–2521 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Eigenfunctions and eigenvalues in superradiance with x–y translational symmetry,” Phys. Lett. A 372, 2787–2801 (2008).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
[CrossRef] [PubMed]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
[CrossRef]

R. Friedberg and S. Hartmann, “Pulse induced radiation in the linear regime,” Opt. Commun. 2, 301–304 (1970).
[CrossRef]

Garrett, W. R.

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

Glauber, R. J.

S. Prasad and R. J. Glauber, “Coherent radiation by a spherical medium of resonant atoms,” Phys. Rev. A 82, 063805 (2010).
[CrossRef]

Hart, R. C.

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

Hartmann, S.

R. Friedberg and S. Hartmann, “Pulse induced radiation in the linear regime,” Opt. Commun. 2, 301–304 (1970).
[CrossRef]

Hartmann, S. R.

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
[CrossRef] [PubMed]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
[CrossRef]

Herman, I. P.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Hughes, I. G.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Keaveney, J.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Krohn, U.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Lamb, W. E.

W. E. Lamb and R. C. Retherford, “Fine structure of the atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[CrossRef]

Lehmberg, R. H.

R. H. Lehmberg, “Radiation from an N-atom system. 1. general formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. 2. emission from a pair of atoms,” Phys. Rev. A 2, 889–896 (1970).
[CrossRef]

MacGillivray, J. C.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Manassah, J. T.

R. Friedberg and J. T. Manassah, “Non-Dicke decay in a small spherical sample with radially varying density,” Phys. Rev A 85, 013834 (2012).
[CrossRef]

J. T. Manassah, “Emission from a slab of resonant two-level atoms induced by a delta-pulse excitation,” Laser Phys. 22, 559 (2012).
[CrossRef]

J. T. Manassah, “Analytic approximate expression for the spectral distribution of the emission from a slab of resonant two-level atoms prepared by an ultrashort δ pulse,” Phys. Rev. A 85, 055801 (2012).
[CrossRef]

J. T. Manassah, “The Purcell–Dicke effect in the emission from a coated small sphere of resonant atoms placed inside a matrix cavity,” Laser Phys. 22, 738–744 (2012).
[CrossRef]

R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic modes of an infinite cylindrical sample of two-level atoms,” J. Math. Phys. 52, 042107 (2011).
[CrossRef]

J. T. Manassah, “The metastable states in the cooperative emission from a system of two-level atoms in a sphere,” Laser Phys. 20, 1397–1403 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state,” Phys. Rev. A 81, 043845 (2010).
[CrossRef]

J. T. Manassah, “Giant cooperative Lamb shift in a density-modulated slab of two-level atoms,” Phys. Lett. A 374, 1985–1988 (2010).
[CrossRef]

J. T. Manassah, “The dynamical cooperative Lamb shift in a system of two-level atoms in a sphere in the scalar photon theory,” Laser Phys. 20, 259–269 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The Eikonal-SVEAS analytic closed forms for single photon superradiance,” Laser Phys. 20, 250–258 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The CLS in an ellipsoid,” Phys. Rev. A 81, 063822 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Analytic expressions for the initial cooperative decay rate and cooperative Lamb shift for a spherical sample of two-level atoms,” Phys. Lett. A 374, 1648–1659 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The metamorphosis in the emission angular profile from an inverted two-level atom system,” Opt. Commun. 282, 3089–3099 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “The dynamical CLS in a system of two-level atoms in a slab-geometry,” Phys. Lett. A 373, 3423–3429 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “Electromagnetic decay modes in a spherical sample of two-level atoms,” Phys. Lett. A 372, 6833–6842 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Time-dependent directionality of cooperative emission after short pulse excitation,” Opt. Commun. 281, 4391–4397 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “The decay dynamics of a slab of two-level atoms excited by an ultrashort resonant pulse,” Opt. Commun. 281, 3755–3761 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Eigenfunctions and eigenvalues in superradiance with x–y translational symmetry,” Phys. Lett. A 372, 2787–2801 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory,” Phys. Lett. A 372, 2514–2521 (2008).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
[CrossRef] [PubMed]

J. T. Manassah, “Statistical quantum electrodynamics of resonant atoms,” Phys. Rep. 101, 359–427(1983).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
[CrossRef]

Payne, M. G.

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

Prasad, S.

S. Prasad and R. J. Glauber, “Coherent radiation by a spherical medium of resonant atoms,” Phys. Rev. A 82, 063805 (2010).
[CrossRef]

Retherford, R. C.

W. E. Lamb and R. C. Retherford, “Fine structure of the atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[CrossRef]

Roehlsberger, R.

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

R. Roehlsberger, “The collective Lamb shift in nuclear γ-ray superradiance,” J. Mod. Opt. 57, 1979–1992 (2010).
[CrossRef]

Ruffer, R.

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

Sahoo, B.

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

Sargsyan, A.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Sarkisyan, D.

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

Schlage, K.

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

Scully, M. O.

A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

A. Svidzinsky and M. O. Scully, “On the evolution of N-atom state prepared by absorption of a single photon,” Opt. Commun. 283, 753–757 (2010).
[CrossRef]

A. Svidzinsky and M. O. Scully, “Evolution of collective N-atom states in single photon superradiance: effect of virtual Lamb shift processes,” Opt. Commun. 282, 2894–2897 (2009).
[CrossRef]

Skribanowitz, N.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Stephen, M. J.

M. J. Stephen, “First order dispersion forces,” J. Chem. Phys. 40, 669–673 (1964).
[CrossRef]

Svidzinsky, A.

A. Svidzinsky, “Nonlocal effects in single photon superradiance,” Phys. Rev. A 85, 013821 (2012).
[CrossRef]

A. Svidzinsky and M. O. Scully, “On the evolution of N-atom state prepared by absorption of a single photon,” Opt. Commun. 283, 753–757 (2010).
[CrossRef]

A. Svidzinsky and M. O. Scully, “Evolution of collective N-atom states in single photon superradiance: effect of virtual Lamb shift processes,” Opt. Commun. 282, 2894–2897 (2009).
[CrossRef]

Svidzinsky, A. A.

A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

Wray, J. E.

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

Yukalov, V. I.

V. I. Yukalov and E. P. Yukalova, “Possibility of superradiance by magnetic nanoclusters,” Laser Phys. Lett. 8, 804–813 (2011); and references to earlier work cited therein.
[CrossRef]

Yukalova, E. P.

V. I. Yukalov and E. P. Yukalova, “Possibility of superradiance by magnetic nanoclusters,” Laser Phys. Lett. 8, 804–813 (2011); and references to earlier work cited therein.
[CrossRef]

Annal. Phys. (1)

R. Friedberg, “Refinement of a formula for decay after weak coherent excitation of a sphere,” Annal. Phys. 325, 345–358 (2010).
[CrossRef]

J. Chem. Phys. (1)

M. J. Stephen, “First order dispersion forces,” J. Chem. Phys. 40, 669–673 (1964).
[CrossRef]

J. Math. Phys. (1)

R. Friedberg and J. T. Manassah, “Electromagnetic modes of an infinite cylindrical sample of two-level atoms,” J. Math. Phys. 52, 042107 (2011).
[CrossRef]

J. Mod. Opt. (1)

R. Roehlsberger, “The collective Lamb shift in nuclear γ-ray superradiance,” J. Mod. Opt. 57, 1979–1992 (2010).
[CrossRef]

J. Phys. B (2)

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Three-photon frequency shift on non-collinear excitation,” J. Phys. B 22, 2211–2222 (1989).
[CrossRef]

R. Friedberg, “Nonradiative transfer of excitation in coherent decay from a Gaussian atomic distribution,” J. Phys. B 44, 175505 (2011).
[CrossRef]

Laser Phys. (5)

R. Friedberg and J. T. Manassah, “The Eikonal-SVEAS analytic closed forms for single photon superradiance,” Laser Phys. 20, 250–258 (2010).
[CrossRef]

J. T. Manassah, “The metastable states in the cooperative emission from a system of two-level atoms in a sphere,” Laser Phys. 20, 1397–1403 (2010).
[CrossRef]

J. T. Manassah, “The dynamical cooperative Lamb shift in a system of two-level atoms in a sphere in the scalar photon theory,” Laser Phys. 20, 259–269 (2010).
[CrossRef]

J. T. Manassah, “Emission from a slab of resonant two-level atoms induced by a delta-pulse excitation,” Laser Phys. 22, 559 (2012).
[CrossRef]

J. T. Manassah, “The Purcell–Dicke effect in the emission from a coated small sphere of resonant atoms placed inside a matrix cavity,” Laser Phys. 22, 738–744 (2012).
[CrossRef]

Laser Phys. Lett. (1)

V. I. Yukalov and E. P. Yukalova, “Possibility of superradiance by magnetic nanoclusters,” Laser Phys. Lett. 8, 804–813 (2011); and references to earlier work cited therein.
[CrossRef]

Opt. Commun. (6)

R. Friedberg and S. Hartmann, “Pulse induced radiation in the linear regime,” Opt. Commun. 2, 301–304 (1970).
[CrossRef]

A. Svidzinsky and M. O. Scully, “On the evolution of N-atom state prepared by absorption of a single photon,” Opt. Commun. 283, 753–757 (2010).
[CrossRef]

A. Svidzinsky and M. O. Scully, “Evolution of collective N-atom states in single photon superradiance: effect of virtual Lamb shift processes,” Opt. Commun. 282, 2894–2897 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “The decay dynamics of a slab of two-level atoms excited by an ultrashort resonant pulse,” Opt. Commun. 281, 3755–3761 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Time-dependent directionality of cooperative emission after short pulse excitation,” Opt. Commun. 281, 4391–4397 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “The metamorphosis in the emission angular profile from an inverted two-level atom system,” Opt. Commun. 282, 3089–3099 (2009).
[CrossRef]

Phys. Lett. A (6)

R. Friedberg and J. T. Manassah, “Electromagnetic decay modes in a spherical sample of two-level atoms,” Phys. Lett. A 372, 6833–6842 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “The dynamical CLS in a system of two-level atoms in a slab-geometry,” Phys. Lett. A 373, 3423–3429 (2009).
[CrossRef]

R. Friedberg and J. T. Manassah, “Eigenfunctions and eigenvalues in superradiance with x–y translational symmetry,” Phys. Lett. A 372, 2787–2801 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory,” Phys. Lett. A 372, 2514–2521 (2008).
[CrossRef]

R. Friedberg and J. T. Manassah, “Analytic expressions for the initial cooperative decay rate and cooperative Lamb shift for a spherical sample of two-level atoms,” Phys. Lett. A 374, 1648–1659 (2010).
[CrossRef]

J. T. Manassah, “Giant cooperative Lamb shift in a density-modulated slab of two-level atoms,” Phys. Lett. A 374, 1985–1988 (2010).
[CrossRef]

Phys. Rep. (2)

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two level atoms,” Phys. Rep. 7, 101–179(1973).
[CrossRef]

J. T. Manassah, “Statistical quantum electrodynamics of resonant atoms,” Phys. Rep. 101, 359–427(1983).
[CrossRef]

Phys. Rev A (1)

R. Friedberg and J. T. Manassah, “Non-Dicke decay in a small spherical sample with radially varying density,” Phys. Rev A 85, 013834 (2012).
[CrossRef]

Phys. Rev. (4)

W. E. Lamb and R. C. Retherford, “Fine structure of the atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[CrossRef]

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[CrossRef]

R. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

D. C. Burnham and R. Y. Chiao, “Coherent resonance fluorescence excited by short light pulses,” Phys. Rev. 188, 667–675 (1969).
[CrossRef]

Phys. Rev. A (10)

J. T. Manassah, “Analytic approximate expression for the spectral distribution of the emission from a slab of resonant two-level atoms prepared by an ultrashort δ pulse,” Phys. Rev. A 85, 055801 (2012).
[CrossRef]

R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
[CrossRef]

A. Svidzinsky, “Nonlocal effects in single photon superradiance,” Phys. Rev. A 85, 013821 (2012).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. 1. general formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. 2. emission from a pair of atoms,” Phys. Rev. A 2, 889–896 (1970).
[CrossRef]

A. A. Svidzinsky, J. T. Chang, and M. O. Scully, “Cooperative spontaneous emission of N atoms: many-body eigenstates, the effect of virtual Lamb shift processes and analogy with radiation of N classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “The CLS in an ellipsoid,” Phys. Rev. A 81, 063822 (2010).
[CrossRef]

R. Friedberg and J. T. Manassah, “Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state,” Phys. Rev. A 81, 043845 (2010).
[CrossRef]

S. Prasad and R. J. Glauber, “Coherent radiation by a spherical medium of resonant atoms,” Phys. Rev. A 82, 063805 (2010).
[CrossRef]

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shift in three-photon resonance,” Phys. Rev. A 39, 93–94 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb shift in an atomic vapor layer of nanometer thickness,” Phys. Rev. Lett. 108, 173601 (2012).
[CrossRef] [PubMed]

W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and M. G. Payne, “Large multiple collective line shifts observed in three-photon excitation of Xe,” Phys. Rev. Lett. 64, 1717–1720 (1990).
[CrossRef] [PubMed]

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Rev. Mod. Phys. (1)

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).
[CrossRef]

Science (1)

R. Roehlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, “Collective Lamb shift in single photon superradiance,” Science 328, 1248–1251 (2010).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. M. Fain, “On the theory of the coherent spontaneous emission,” Sov. Phys. JETP 9, 562–565 (1959).

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Figures (25)

Figure 1
Figure 1

(a) The reduced CDR and (b) the reduced CLS are plotted as functions of u 0 = k 0 z 0 . The atomic density is uniform. Slab geometry. The initial state of the atoms is exp i k 0 z .

Figure 2
Figure 2

(a) The reduced CDR and (b) the reduced CLS are plotted as functions of β = k 1 k 0 / k 0 . The atomic density is uniform. Slab geometry. The initial state of the system is exp i k 1 z . u 0 = 100 π .

Figure 3
Figure 3

(a) The reduced CDR and (b) the reduced CLS are plotted as functions of u 0 . The atomic density is uniform. Slab geometry. The initial state of the system is exp i k 1 z . β u 0 = π / 2 .

Figure 4
Figure 4

(a) The reduced CDR and (b) the reduced CLS are plotted as functions of u 0 . The atomic density is uniform. Slab geometry. The initial state of the system is exp i k 1 z . β u 0 = π .

Figure 5
Figure 5

(a) Γ mod C D R / Γ u n i f C D R and (b) Δ Ω mod C L S / Δ Ω L o r e n t z are plotted as functions of α. The atomic density is modulated; depth of modulation a = 1 . Slab geometry. m = 100 . The initial state of the system is exp i k 0 z .

Figure 6
Figure 6

(a) Γ mod C D R / Γ u n i f C D R and (b) Δ Ω mod C L S / Δ Ω L o r e n t z are plotted as functions of α. The atomic density is modulated, depth of modulation: a = 1 . Slab geometry. m = 300 . The initial state of the system is exp ( i k 0 z ) .

Figure 7
Figure 7

(a) Re Σ ̃ and (b) Im Σ ̃ are plotted as functions of u 0 . Scalar theory. Sphere geometry. The atomic density is uniform. The initial state of the system is uniform.

Figure 8
Figure 8

(a) Re Σ ̃ and (b) Im Σ ̃ are plotted as functions of u 0 . Scalar theory. Sphere geometry. The atomic density is uniform. The initial state of the system is exp i k 0 z .

Figure 9
Figure 9

(a) Re Σ ̃ and (b) Im Σ ̃ are plotted as functions of β. Scalar theory. Sphere geometry. The atomic density is uniform. The initial state of the system is exp i k 1 z . k 1 = k 0 ( 1 + β ) . u 0 = 100 π .

Figure 10
Figure 10

(a) Re Σ ̃ and (b) Im Σ ̃ are plotted as functions of u ̄ . Scalar theory. Sphere geometry. The atomic density is Gaussian. The initial state of the system is uniform.

Figure 11
Figure 11

(a) Re Σ ̃ and (b) Im Σ ̃ are plotted as functions of u ̄ . Scalar theory. Sphere geometry. The atomic density is Gaussian. The initial state of the system is exp i k 0 z .

Figure 12
Figure 12

(a) Re Σ ̃ and (b) Im Σ ̃ Σ ̃ C o u l are plotted as functions of u 0 . Vector theory. Sphere geometry. The atomic density is uniform. The initial state of the system is exp i k 0 z . (The dashed traces are those of the scalar theory shown in Fig. 8.)

Figure 13
Figure 13

(a) Re Σ ̃ and (b) Im Σ ̃ Σ ̃ C o u l are plotted as functions of u ̄ . Vector theory. Sphere geometry. The atomic density is Gaussian. The initial state of the system is exp i k 0 z . (The dashed traces are those of the scalar theory shown in Fig. 11.)

Figure 14
Figure 14

The normalized CDR is plotted as function of u ̄ l for fixed u ̄ t = 10 . Ellipsoid geometry. Scalar theory.

Figure 15
Figure 15

The normalized CLS is plotted as function of u ̄ l for fixed u ̄ t = 10 . o are the values numerically computed by using Eq. (5.6); l are the values given by the approximate expression, Eq. (5.4); and h are the values given by the approximate expression, Eq. (5.5). Ellipsoid geometry. Scalar theory.

Figure 16
Figure 16

(a) Re Σ ̃ S α = 1 , (b) Im Σ ̃ S α = 1 are plotted as functions of u 0 . Scalar theory. Infinite cylinder geometry. (Recall that α = 1 also means k = 0 .)

Figure 17
Figure 17

Im Σ ̃ S is plotted as function of u 0 , for α = 0 . 02 . Scalar theory. Infinite cylinder geometry.

Figure 18
Figure 18

Im Σ ̃ S is plotted as function of | α | , α < 0 , for u 0 = 12 π . Scalar theory. Infinite cylinder geometry.

Figure 19
Figure 19

Im Σ ̃ T E α = 1 is plotted as function of u 0 . Vector theory. Infinite cylinder geometry.

Figure 20
Figure 20

Im Σ ̃ T E is plotted as function of u 0 for α = 0 . 25 . Vector theory. Infinite cylinder geometry.

Figure 21
Figure 21

Comparison of the numerical exact results (solid curve) for the forward emission spectral distribution with those of the SVEA expressions (dashed curve). Slab geometry. u 0 = k 0 z 0 = 12 . 25 π , Γ ̃ 2 = 2 . 33 / 4 .

Figure 22
Figure 22

Comparison of the numerical exact results (solid curve) for the forward emission spectral distribution with those of the SVEA expressions (dashed curve). Slab geometry. u 0 = k 0 z 0 = 100 . 25 π , Γ ̃ 2 = 2 . 33 / 4 .

Figure 23
Figure 23

Comparison of the numerical exact results (solid curve) for the forward emission spectral distribution with those of the MSVEA expressions (dotted curve). [Note: There is no printing error, the two traces truly merge]. u 0 = k 0 z 0 = 12 . 25 π , Γ ̃ 2 = 2 . 33 / 4 . Slab geometry.

Figure 24
Figure 24

Comparison of the numerical exact results (solid line) for the forward emission spectral distribution with those of the MSVEA expressions (dotted line). [Note: There is no printing error, the two traces truly merge]. u 0 = k 0 z 0 = 100 . 25 π , Γ ̃ 2 = 2 . 33 / 4 . Slab geometry.

Figure 25
Figure 25

The forward emission spectral distribution for large normalized frequencies for the parameters of Fig. 24. Slab geometry.

Tables (1)

Tables Icon

Table 1. N-Atom Effects

Equations (161)

Equations on this page are rendered with MathJax. Learn more.

d d t ln a | P ( a ) | 2 = 2 Re a P ( a ) P ̇ ( a ) a | P ( a ) | 2 .
2 Re ( Σ ( t = 0 ) ) = Initial Cooperative Decay Rate,
Im ( Σ ( t = 0 ) ) = Initial Frequency Shift,
Σ = a P ( a ) P ̇ ( a ) a | P ( a ) | 2 .
P ̇ j ( r , t ) = N 2 k 0 3 ħ d 3 r ρ ( r ) i = 1 3 d 3 r G i , j ( r r ) ρ ( r ) P i ( r , t ) ,
G ( ) = i k 0 3 exp ( i k 0 ) I 3 ˆ ˆ 3 ( 1 + i k 0 ) ( I ˆ ˆ ) k 0 2 ,
b ̇ r , t = N γ 1 2 d 3 r ρ r d 3 r G r r ρ r b r , t
G = exp i k 0 i k 0 .
2 k 0 3 ħ = 3 4 γ 1 = 3 2 γ 1 2 ,
Σ S 0 = N γ 1 2 d 3 r ρ r d 3 r ρ r d 3 r d 3 r b r ρ r G r r ρ r b r
Σ V 0 = N 2 k 0 3 ħ d 3 r d 3 r ρ r ρ r d 3 r d 3 r i , j 3 P i r ρ r G i , j r r ρ r P j r
G ̃ z z = 0 d P 2 π P exp i k 0 P 2 + z z 2 i k 0 P 2 + z z 2 = 2 π k 0 2 exp i k 0 | z z | .
Δ Ω L o r e n t z = n π γ 1 k 0 3
4 3 2 k 0 3 ħ = γ 1 .
Γ C D R = 3 4 n π γ 1 k 0 3 1 cos ( 4 u 0 ) + 8 u 0 2 2 u 0 ,
Δ Ω C L S = 3 4 n π γ 1 k 0 3 sin 4 u 0 4 u 0 4 u 0 .
Γ C D R = 3 4 n π γ 1 k 0 3 8 u 0 16 3 u 0 3 + O u 0 4 ,
Δ Ω C L S = 3 4 n π γ 1 k 0 3 8 3 u 0 2 + O u 0 4 ,
Γ C D R = 3 4 n π γ 1 k 0 3 4 u 0 + O 1 u 0 ,
Δ Ω C L S = 3 4 n π γ 1 k 0 3 1 + O 1 u 0 .
Γ C D R = 3 4 n π γ 1 k 0 3 Γ ¯ C D R a n d Δ Ω C L S = 3 4 n π γ 1 k 0 3 Δ Ω ¯ C L S .
n ( z ) = n 0 exp ( z 2 / ( 2 l ̄ 2 ) ) ,
Γ C D R = 3 n γ 1 π k 0 3 u ̄ ( 1 + exp ( 4 u ̄ 2 ) ) ,
Δ Ω C L S = 3 n γ 1 π k 0 3 u ̄ F ( 2 u ̄ ) ,
F x = exp x 2 0 x exp y 2 d y .
lim x 1 F ( x ) = x + O ( x 3 ) ,
lim x 1 F x = 1 2 x + O x 3 .
β u 0 = k 1 k 0 z 0 ,
β u 0 = k 1 + k 0 z 0 = 2 u 0 + β u 0 ,
Γ C D R = 3 4 n π γ 1 k 0 3 2 β 2 u 0 1 cos 2 β u 0 + 2 β 2 u 0 1 cos 2 β u 0 ,
Δ Ω C L S = 3 4 n π γ 1 k 0 3 1 β 2 u 0 2 β u 0 sin 2 β u 0 + 1 β 2 u 0 2 β u 0 + sin 2 β u 0 .
Γ C D R = 3 4 n π γ 1 k 0 3 8 u 0 8 3 2 + 2 β + β 2 u 0 3 + O u 0 4 ,
Δ Ω C L S = 3 4 n π γ 1 k 0 3 8 3 u 0 2 + O u 0 4 .
Γ C D R = 3 4 n π γ 1 k 0 3 Γ ¯ C D R and Δ Ω C L S = 3 4 n π γ 1 k 0 3 Δ Ω ¯ C L S
ρ ( r ) 1 a cos ( Q z ) ,
Σ mod = 3 4 n π γ 1 k 0 3 1 4 α ( 1 4 α 2 ) 2 m π ( 1 exp ( i 4 α m π ) ) ( 1 4 α 2 ( 1 a ) ) 2 + 4 i α m π ( 1 2 ( 4 + a 2 ) α 2 + 8 ( 2 + a 2 ) α 4 ) + 8 α 2 ( m π ) 2 ( 1 4 α 2 ) 2 ,
Γ mod C D R α = 1 / 2 Γ u n i f C D R 1 + a 2 4 ,
Δ Ω mod C L S α = 1 / 2 Δ Ω L o r e n t z 3 4 1 + a a 2 8 .
Σ mod 3 4 n π γ 1 k 0 3 a 2 1 exp ( 4 i m π Δ α ) i 4 m π Δ α ( 16 m π Δ α ) ( Δ α ( 1 Δ α ) ) ,
Re Σ mod 3 4 n π γ 1 k 0 3 a 2 m π 4 sin 2 2 m π Δ α 2 m π Δ α 2
Im Σ mod 3 4 n π γ 1 k 0 3 a 2 m π 2 sin 4 m π Δ α 4 m π Δ α 4 m π Δ α 2 .
Δ α ( max s h i f t ) m 1 4 m ,
Δ Ω G i a n t S h i f t s Δ Ω L o r e n t z m ± 3 8 a 2 m .
m ( max s h i f t ) Δ α β 4 π | Δ α | ,
Δ Ω G i a n t S h i f t s Δ Ω L o r e n t z Δ α 3 4 sin ( β ) β 8 β a 2 Δ α 0 . 114 Δ α a 2 .
I f = d 3 r ρ r d 3 r ρ r f ,
ρ ( r ) = 1 for  r < R 0 for  r > R ,
I f = π 2 3 0 2 R d 2 R 2 4 R + 2 f .
ρ ( r ) = exp 1 2 r l ̄ 2 ,
I f = 4 π 5 / 2 l ̄ 3 0 d 2 exp 2 4 l ̄ 2 f .
Σ ̃ = 3 2 u 0 6 [ 3 + 3 u 0 2 + 2 i u 0 3 3 ( 1 i u 0 ) 2 exp ( i 2 u 0 ) ] ,
Re ( Σ ̃ ) = 9 [ u 0 cos ( u 0 ) sin ( u 0 ) ] 2 u 0 6 = 3 j 1 ( u 0 ) u 0 2 ,
Σ ̃ = 3 256 u 0 6 3 24 u 0 2 64 i u 0 3 + 96 u 0 4 + 3 1 4 i u 0 exp i 4 u 0 .
Re ( Σ ̃ ) 1 2 u 0 2 5 + O ( u 0 4 ) , Im ( Σ ̃ ) 6 5 u 0 + 24 u 0 35 + O ( u 0 3 ) .
Re ( Σ ̃ ) 9 8 u 0 2 , Im ( Σ ̃ ) 3 4 u 0 3 9 64 cos ( 4 u 0 ) u 0 5 .
Σ ̃ = 3 i 8 β 4 β + 1 β + 2 4 u 0 6 3 i exp i 2 u 0 sin 2 β + 1 u 0 β 4 6 u 0 i + 4 β 3 6 u 0 i + 4 β 2 8 u 0 3 i + 16 β u 0 i 8 i + 6 β + 1 exp i 2 u 0 cos 2 β + 1 u 0 β 4 u 0 + 4 β 3 u 0 + β 2 8 u 0 2 i + β 8 u 0 4 i 4 i + 4 β + 1 2 β 6 u 0 3 + 12 β 5 u 0 3 + 3 β 4 u 0 2 8 u 0 + i + 4 β 3 u 0 2 4 u 0 + 3 i + 3 i β 2 4 u 0 2 + 1 + 6 i β + 6 i
Σ ̃ 9 8 u 0 2 i 3 4 u 0 3 ,
Σ ̃ 0 . 545 u 0 2 i 0 . 682 u 0 2 .
Σ ̃ = exp ( u ̄ 2 ) + i π u ̄ [ 2 u ̄ F ( u ̄ ) 1 ] ,
Σ ̃ = 1 4 u ̄ 2 ( 1 exp ( 4 u ̄ 2 ) ) i F ( 2 u ̄ ) 2 π u ̄ 2 ,
G i j C o u l = i δ i j 3 ˆ i ˆ j 3
Re ( Σ ̃ ) = 27 512 u 0 6 7 + 40 u 0 2 + 32 u 0 4 + 7 cos ( 4 u 0 ) 4 u 0 sin ( 4 u 0 ) + 8 ( 1 4 u 0 2 ) [ γ C i ( 4 u 0 ) + ln ( 4 u 0 ) ] , Im ( Σ ̃ ) = 21 8 u 0 3 + 27 512 u 0 6 7 sin ( 4 u 0 ) 4 u 0 cos ( 4 u 0 ) + 8 ( 1 4 u 0 2 ) S i ( 4 u 0 ) ,
Σ ̃ C o u l . = i 3 64 u 0 6 32 u 0 3 18 u 0 cos ( 2 u 0 ) 9 sin ( 2 u 0 ) + ( 18 36 u 0 2 ) S i ( 2 u 0 ) ,
Re ( Σ ̃ ) = 1 2 5 u 0 2 + O ( u 0 4 ) ,
Im ( Σ ̃ ) = 33 25 1 u 0 + 828 1225 u 0 + O ( u 0 3 ) ,
Im ( Σ ̃ C o u l ) = 3 25 1 u 0 + 9 1225 u 0 + O ( u 0 3 ) .
Re Σ ̃ = 3 32 u ̄ 6 4 u ̄ 4 2 u ̄ 2 + 1 exp 4 u ̄ 2 4 u ̄ 4 + 2 u ̄ 2 + 1 ,
Im Σ ̃ = 1 32 π u ̄ 6 8 u ̄ 3 + 12 u ̄ 6 4 u ̄ 4 + 2 u ̄ 2 + 1 F 2 u ̄ ,
Σ ̃ C o u l = i 1 8 π u ̄ 6 2 u ̄ 3 3 u ̄ + 3 F u ̄ .
Re ( Σ ̃ ) = 1 2 u ̄ 2 + O ( u ̄ 4 ) ,
Im ( Σ ̃ ) = 11 10 π u ̄ + 92 35 π u ̄ + O ( u ̄ 3 ) ,
Im Σ ̃ C o u l = 1 10 π u ̄ + 1 35 π u ̄ + O u ̄ 3 .
ρ ̃ r = exp z 2 2 l l 2 2 π l l 1 / 2 exp x 2 + y 2 2 l t 2 2 π l t ,
Γ C D R = N γ 1 2 F u t 2 u t 2 u l 2 exp 4 u l 2 F 2 u l 2 u t 2 u t 2 u l 2 u t 2 u l 2 .
Δ Ω C L S = N γ 1 2 1 4 π 1 / 2 u ̄ l u ̄ t 2 0 d ξ exp 1 u ̄ l 2 1 u ̄ t 2 ξ 2 4 cos ξ ξ d σ exp σ 2 4 u ̄ t 2 cos σ .
Δ Ω C L S 1 u ̄ l u t 2 N γ 1 2 1 64 π 1 / 2 u ̄ l u ̄ t 2 2 u ̄ l 2 u ̄ t 2 1 .
Δ Ω C L S u ̄ l u ̄ t 2 1 N γ 1 2 1 4 π ln ( u l u t 2 ) 1 2 γ u l ,
Σ = N γ 1 2 1 2 i u ̄ t u ̄ l exp ( u ̄ t 2 ) 0 d ξ exp 1 u ̄ t 2 1 u ̄ l 2 ξ 2 4 cos ( ξ ) Erfc ξ 2 u ̄ t i u ̄ t ,
G ( r r ) = 1 2 k 0 m = exp ( i m ( ϕ ϕ ) ) d k z exp ( i k z ( z z ) ) J m ( k T ρ < ) H m ( 1 ) ( k T ρ > ) ,
G i , j ( ) = G ( ) δ i , j + 1 k 0 2 i j G ( ) ,
Σ S = n γ 1 2 4 π 2 k 0 3 Σ ̃ S .
Re Σ ̃ S α > 0 = 1 α J 1 k 0 R α 2 ,
Im Σ ̃ S α > 0 = 1 α N 1 k 0 R α J 1 k 0 R α + 1 π .
Re Σ S . k = 0 n γ 1 2 π 2 k 0 3 k 0 R 4 ,
Im Σ S k = 0 n γ 1 2 π 2 k 0 3 1 + 4 γ 4 ln 2 + 4 ln ( k 0 R ) k 0 R 2 ,
Re Σ ̃ S α < 0 = 0 ,
Im Σ ̃ S α < 0 = 1 | α | π 1 2 K 1 k 0 R | α | I 1 k 0 R | α | .
Re Σ ̃ T M α > 0 = J 1 k 0 R α 2 ,
Im Σ ̃ T M α > 0 = N 1 k 0 R α J 1 k 0 R α + 1 π .
Re Σ ̃ T M α < 0 = 0 ,
Im Σ ̃ T M α < 0 = 1 π 1 2 K 1 k 0 R | α | I 1 k 0 R | α | .
Re Σ ̃ T E α > 0 = 1 α 1 α 2 J 1 k 0 R α 2 ,
Im Σ ̃ T E α > 0 = 1 α 1 α 2 N 1 k 0 R α J 1 k 0 R α + 1 π .
Re Σ ̃ T E α < 0 = 0 ,
Im Σ ̃ T E α < 0 = 1 | α | π 1 2 1 + | α | 2 K 1 k 0 R | α | I 1 k 0 R | α | .
b ̇ ( z , t ) i Δ Ω L o r e n t z b ( z , t ) + Γ T b ( z , t ) = C k 0 2 z 0 z 0 d z exp ( i k 0 | z z | ) b ( z , t ) ,
E z , t = A z 0 z 0 d z exp i k 0 | z z | b z , t
E ( z 0 , t ) = A exp ( i k 0 z 0 ) z 0 z 0 d z exp ( i k 0 z ) b ( z , t ) ,
E z 0 , t = A exp i k 0 z 0 z 0 z 0 d z exp i k 0 z b z , t .
b z , t = exp Γ 2 i Δ Ω L o r e n t z t exp i k 0 z B z , t ,
B ̇ z , t = C k 0 2 z 0 z B z , t d z + exp 2 i k 0 z z z 0 B z , t exp 2 i k 0 z d z .
2 B Z , T Z T = u 0 B Z , T
2 U 2 + 1 U U B U , R R 2 U 2 2 R 2 + 1 R R B U , R = 2 u 0 B U , R .
B U , R = A m R m Q m U .
d 2 Q 0 U d U 2 + 1 U d Q 0 U d U + 2 u 0 Q 0 U = 0 .
B Z , T = B 0 J 0 2 u 0 Z + 1 T .
E f S V E A ( T ) E f S V E A ( 0 ) = exp ( ( Γ ̃ 2 i Δ Ω ̃ L o r e n t z ) T ) 1 1 d Z J 0 ( 2 u 0 ( Z + 1 ) T ) 1 1 d Z = exp ( ( Γ ̃ 2 i Δ Ω ̃ L o r e n t z ) T ) J 1 4 u 0 T u 0 T ,
a ̃ f S V E A ( Ω ̃ ) = 1 2 0 d T exp ( i Ω ̃ T ) exp ( ( Γ ̃ 2 i Δ Ω ̃ L o r e n t z ) T ) J 1 4 u 0 T u 0 T = 1 u 0 ( 1 exp ( i u 0 / Δ S V E A ) ) ,
B z , t t t = 0 = i Δ Ω L o r e n t z Γ T B z , 0 C k 0 2 z 0 z 0 d z exp i k 0 | z z | exp i k 0 z z B z , 0 .
B z , t B z , t t t = 0 ¯ = i Δ Ω L o r e n t z Γ T C k 0 4 z 0 z 0 z 0 d z z 0 z 0 d z exp i k 0 | z z | exp i k 0 z z .
1 1 d Z 1 1 d Z exp i u 0 | Z Z | exp i u 0 Z Z = 2 exp 4 i u 0 1 4 u 0 2 i u 0 ,
Γ ̃ C D R = 1 8 u 0 1 cos 4 u 0 + 8 u 0 2 ,
Δ Ω ̃ C L S = 1 4 1 sin ( 4 u 0 ) 4 u 0 = 3 4 Δ Ω ̃ L o r e n t z 1 sin ( 4 u 0 ) 4 u 0 .
d d T J 1 2 4 u 0 T u 0 T T = 0 = u 0 ,
Δ Ω ̃ M = Δ Ω ̃ L o r e n t z + Δ Ω ̃ C L S = Δ Ω ̃ L o r e n t z 1 3 4 1 sin ( 4 u 0 ) 4 u 0 = Δ Ω ̃ L o r e n t z 1 4 + 3 16 sin ( 4 u 0 ) u 0 ,
a ̃ f M S V E A ( Ω ) = 1 u 0 ( 1 exp ( i u 0 / Δ M S V E A ) ) ,
H f = 1 8 π d 3 r ( Π 2 + ( × A ) 2 ) ,
A ( r , t ) = k ε ˆ k 2 π ħ c V k ε ˆ ( a k , ε ˆ exp ( i k r i ω t ) + h . c . ) ,
Π ( r , t ) = k ε ˆ k 2 π ħ c k V ε ˆ ( a k , ε ˆ exp ( i k r i ω t ) + h . c . ) ,
A i r , t , Π j r , t = δ i , j δ 3 r r ,
H f = k ε ˆ k 1 2 a k , ε ˆ a k , ε ˆ + + a k , ε ˆ + a k , ε ˆ ħ ω .
1 i ħ [ A , H f ] = c Π ,
Π = 1 c A ̇
H = H f + H a t + H d i p d i p + H r a d = H f + H a t + H I .
τ α , i = | i 0 | α , τ α , i = | 0 i | α .
H d i p d i p = α < β 1 r α , β 3 ( p α p β 3 p α r ˆ α , β p β r ˆ α , β ) ,
p α = ( τ α , 1 x ˆ + τ α , 2 y ˆ + τ α , 3 z ˆ ) exp ( i ω 0 t ) + h . c . ,
H d i p d i p = 1 2 α β i , j ( τ α , i τ β , j + τ α , i τ β , j j ) W i , j ( r α , β ) ,
W i , j r = 2 r 3 δ i , j 3 r ˆ i r ˆ j .
H r a d = 1 c J r , t A r , t d 3 r = 1 c α p ̇ α A r α , t = α i ω 0 c i τ α , i exp i ω 0 t τ α , i exp i ω 0 t A i r α , t .
g 0 = 2 π ħ c 2 V c ,
H r a d = α k , ε ˆ k g 0 ω i ω 0 i τ α , i exp i ω 0 t + h . c . ε ˆ i a k , ε ˆ exp i k r α i ω t + h . c . .
ψ ( t ) = α , i b α , i ( t ) τ α , i | g d + k , ε ˆ B k , ε ˆ ( t ) a k , ε ˆ + | g d + k , ε ˆ α , i β , j B k , ε ˆ , α , i , β , j a k , ε ˆ + τ α , i τ β , j | g d + higher order terms,
b ̇ β , j ( t ) 1 i ħ α , i b α , i ( t ) W i , j ( r α , β ) = 1 i ħ k , ε ˆ g 0 ω ( i ω 0 ) B k , ε ˆ ( t ) exp ( i ω 0 t ) exp ( i k r β i ω t ) ε j + α , i B k , ε ˆ , α , i , β , j ( t ) exp ( i ω 0 t ) exp ( i k r α i ω t ) ε i
β | b β , j 0 | 2 = 1 ;
B ̇ k , ε ˆ t = 1 i ħ g 0 ω i ω 0 α , i b α , i t exp i ω 0 t exp i k r α + i ω t ε i + 2 photon terms
B k , ε ˆ 0 = 0 ;
B ̇ k , ε ˆ , α , i , β , j t = 1 i ħ g 0 ω i ω 0 b α , i t exp i ω 0 t exp i k r β + i ω t ε j + α β + 2 photon terms
B k , ε ˆ , α , i , β , j 0 = 0 .
b ̇ β , j ( t ) 1 i ħ α β , i b α , i ( t ) W i , j ( r α , β ) + α β 0 t b α , i ( t ) G α , i , β , j ( t t ) d t γ 1 2 b β , j ( t ) ,
G α , i , β , j ( u ) = 1 i ħ 2 k , ε ˆ k g 0 2 ω ( ω 0 2 ) exp ( i ω u ) ε i ε j { exp ( i k ( r β r α ) + i ω 0 u ) + c . c . } ,
ε ˆ k ε i ε j = δ i , j k ˆ i k ˆ j .
k V 2 π 3 d Ω k 2 d k .
C i , j ζ , r ˆ = δ i , j k ˆ i k ˆ j a z = T ζ δ i , j + Y ζ r ˆ i r ˆ j
3 T + Y = i C i , i = 3 1 a z = 2 ,
T + Y = i , j C i , j r ˆ i r ˆ j = 1 ( k ˆ r ˆ ) 2 a z = 1 ζ 2 ,
C i , j = 1 2 1 + ζ 2 δ i , j + 1 2 1 3 ζ 2 r ˆ i r ˆ j .
1 1 d x exp i a x = 2 sin ( a ) a ,
1 1 d x x 2 exp i a x = 2 sin a a + 4 cos ( a ) a 2 4 sin ( a ) a 3 ,
G i , j ( u ) = 2 ω 0 2 2 π ħ c 3 0 d ω ω exp ( i ω u ) sin ( ω r c ) ( ω r c ) cos ( ω 0 u ) [ δ i , j r ˆ i r ˆ j ] + cos ( ω r c ) ( ω r c ) 2 sin ( ω r c ) ( ω r c ) 3 cos ( ω 0 u ) [ δ i , j 3 r ˆ i r ˆ j ] ,
0 d ω ω exp ( i ω u ) sin ω r c ω r c = c 2 r i 1 ε + i u i r / c 1 ε + i u + i r / c ,
0 d ω ω exp i ω u cos ω r c ω r c 2 sin ω r c ω r c 3 = u tanh 1 r u c r c r c 3 .
G i , j ( r ) = 2 ω 0 2 2 π ħ c 3 c cos ( ω 0 u ) 2 i r 1 ε + i u i r c 1 ε + i u + i r c [ δ i , j r ˆ i r ˆ j ] + cos ( ω 0 u ) u tanh 1 ( r u c ) r c ( r c ) 3 [ δ i , j 3 r ˆ i r ˆ j ] .
1 2 ln u + r / c u r / c
1 2 ln u + r / c ( r / c u ) exp ( i π ) = 1 2 ln u + r / c ( r / c u ) + i π .
0 d u cos ω 0 u 1 ε + i u i r / c 1 ε + i u + i r / c = d u cos ω 0 u 1 ε + i u i r / c = π exp i ω 0 r c
0 d u cos ω 0 u u tanh 1 r u c r c = π 2 i ω 0 2 exp i ω 0 r c 1 + r ω 0 c exp i ω 0 r c .
b ̇ β , j ( t ) α β , i b α , i ( t ) K α , i , β , j γ 1 2 b β , j ( t ) ,
K α , i , β , j = 2 i ħ exp ( i k 0 r α , β ) 1 r α , β 3 i k 0 r α , β 2 ( δ i , j 3 ( r ˆ α , β ) i ( r ˆ α , β ) j ) k 0 2 r α , β ( δ i , j ( r ˆ α , β ) i ( r ˆ α , β ) j ) .
b ̇ j ( r , t ) = i N 2 i V ħ d 3 r ρ ( r ) exp ( i k 0 R ) 1 R 3 i k 0 R 2 ( δ i , j 3 R ˆ i R ˆ j ) k 0 2 R ( δ i , j R ˆ i R ˆ j ) b i ( r , t ) γ 1 2 b j ( r , t ) .
b ̇ j ( r , t ) = i N 2 i V ħ d 3 r ρ ( r ) exp ( i k 0 R ) 1 R 3 i k 0 R 2 ( δ i , j 3 R ˆ i R ˆ j ) k 0 2 R ( δ i , j R ˆ i R ˆ j ) b i ( r , t R / c ) γ 1 2 b j ( r , t ) .

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